Simulating groundstate and dynamical quantum phase transitions on a superconducting quantum computer

The phenomena of quantum criticality underlie many novel collective phenomena found in condensed matter systems. They present a challenge for classical and quantum simulation, in part because of diverging correlation lengths and consequently strong finite-size effects. Tensor network techniques that work directly in the thermodynamic limit can negotiate some of these difficulties. Here, we optimise a translationally invariant, sequential quantum circuit on a superconducting quantum device to simulate the groundstate of the quantum Ising model through its quantum critical point. We further demonstrate how the dynamical quantum critical point found in quenches of this model across its quantum critical point can be simulated. Our approach avoids finite-size scaling effects by using sequential quantum circuits inspired by infinite matrix product states. We provide efficient circuits and a variety of error mitigation strategies to implement, optimise and time-evolve these states.


Supplementary Note -Order of Trotterisation
One of the key refinement parameters in time-evolving circuit states is the order of Trotterisation. Supplementary Fig.1 illustrates the different transfer matrices that result from a second-order Trotterisation and from a first-order Trotterisation that we have simplified using a property of translationally invariant states. Because of the projection back to translationally invariant states, this update is in fact effective to higher order in dt. This can be demonstrated as follows: the time-dependent variational principle equations for evolving a translationally invariant state with just the even-or odd-bond parts of the Hamiltonian are identical to evolving using the full Hamiltonian divided by two [1]. Our algorithms are equivalent to a discretisation of the time-dependent variational principle [2], since the latter continuously projects the evolved state back to the state on the MPS manifold that optimises fidelity and our algorithms involve an explicit optimisation of fidelity at discrete time-steps. The results presented below and in the main paper are exclusively for this simplified first-order Trotterisation in order to avoid the deeper circuit required for higher-order Trotterisation.

Supplementary Note -Cost Functions for Time-evolution
As discussed in Section II C of the main text, finding an appropriate cost-function for time-evolution quantum circuit iMPS involves a subtle trade-off between the analytical approximation to the eigenvalue of the transfer matrix -determined by order of Trotterisation, order of power method used, and accuracy of approximations to the transfer matrix fixed-points -and the infidelities of representing this circuit on a real device. Here we give results for a number of different cost functions, the dynamics that they predict in the absence of circuit noise and their realisation on the Rainbow device.
Supplementary Figure 1: Time-evolution Transfer Matrix a) The overlap | ψ(U )|e Hdt |ψ(U ) | 2 , calculated with a second order Trotterisation. Here W even = e iHedt and W odd = e iHodt/2 . This overlap is formally an infinite width and depth circuit. b) An expression for the overlap with one half timestep as used here. Mapping back to a translationally-invarient variational manifold increase sthe effective order of the time-integration.

Approximating the right fixed point with |0 0|
First we show in Figs. 2 and 3 the result of using an identity approximation to the left eigenvalue and a simple |0 0| approximation to the right eigenvalue. As shown in 2, this performs very poorly when using the ratio C 2 /C 1 to estimate the principle eigenvalue of the transfer matrix. This is true even when the circuits are calculated in simulations in the absence of noise. This is apparently due to the low order of the power method, despite the good approximation to the left fixed point. When the same method is used to calculate C 5 /C 4 the approximation performs well in simulations without noise.
Curiously, simply using C 2 as an approximation to λ 2 performs much better. Supplementary Fig.3 shows that, in the absence of circuit errors, this circuit does a passable job of capturing the dynamical quantum phase transition in the quench dynamics of the quantum Ising model. However, when implemented on the Rainbow device, the optimum measured cost function is not at the correct updated values -even after we carry out our rescaling to account for depolarisation error.

Improved approximations to the right fixed-point.
While in principle, a good approximation to the principal eigenvalue of the transfer matrix can be found using a good initial approximation to only either the left or right fixed-points of the transfer matrix. We find in practice that we get much better results if our approximations to both left and right fixed points are good. Supplementary Figs.4 and ?? show the results of progressive improvements to the approximation.
In Supplementary Fig. 4 we show the results of using the same circuit as in Fig. 3 in the main paper, but measuring the probability of |0 ⊗6 at the output rather than post-selecting the two top right qubits on |0 . This amounts to using U and U to construct an approximation to the right fixed-point. The resulting circuit does a passable job of capturing the dynamical quantum phase transition in the absence of circuit errors, and the cost function is apparently tracked rather well when implemented on the Rainbow device. However, the measured optimum value of this cost function is not correct. Post-selecting on the top right qubits corrects these deficiencies. This procedure factors out the U -dependent norm of the approximation Supplementary Figure 2: Time-Evolution Circuits and Results: approximating the right fixed point with |0 0| Ratio of C 2 /C 1 ≈ λ. a) Time-evolution circuit: This circuit approximates the left fixed point as I and the right fixed point as |0 0|. The probability of measuring |0 ⊗N at the output is divided by the appropriate Loschmidt echo and the ratio of the output with two copies of the transfer matrix divided by the output for a circuit with one copy of the transfer matrix. This gives an approximation to λ -the principal eigenvalue of the transfer matrix. b) Dynamical Quantum Phase Transition in the Quantum Ising Model: Exact analytical results of the time evolution and a simulation using direct stochastic optimisation of the circuit in a) in the absence of noise. Taking the ratio of C 2 /C 1 approximated with the circuit shown in a) does not reproduce the correct time evolution. Calculating C 5 /C 4 using the same approximation for the right fixed point does a much better job at capturing the time-evolution suggesting that it is the low order of the power method that leads to the failure. c)Cost-function evaluated on Google's Rainbow Device: The cost function evaluated along a linear interpolation from U = U through the optimal value of U as in Fig.3 of the main text. Though this displays a peak at the optimum value it is not the global optimum of the circuit. to the right fixed-point, giving better performance both under simulation in the absence of error and when implemented on the Rainbow device.
Finally, we report on two further approximations to the right fixed point that do not require postselection, and may perform even better than the circuit reported in the main paper. Unfortunately, it was not possible to test these circuits on the Rainbow device as they were discovered after the machine was taken offline. These correspond to approximating the right fixed-point by that obtained when overlapping the state described by U with itself. This is a good approximation since evolving the state described by U backwards in time ought to recover the state described by U . The two approximation shown in Supplementary Fig.?? correspond to using the fixed point determined by solving the fixed-point equations of Fig.4c in the main paper for V (U ) and an approximation to this obtained by simply overlapping U with itself to construct an approximation to the right fixed point. Both approximations do an excellent job of capturing the evolution in the absence of noise and we believe will provide a good starting point for future investigation.